Effective finite termination procedures in interior-point methods for linear programming
Williams, Pamela Joy
Tapia, Richard A.; El-Bakry, Amr S.
Doctor of Philosophy
Due to the structure of the solution set, an exact solution to a linear program cannot be computed by an interior-point algorithm without adding features, such as finite termination procedures, to the algorithm. Finite termination procedures attempt to compute an exact solution in a finite number of steps. The addition of a finite termination procedure enables interior-point algorithms to generate highly accurate solutions for problems in which the ill-conditioning of the Jacobian in the neighborhood of the solution currently precludes such accuracy. The main ingredients of finite termination are activating the procedure, predicting the optimal partition, formulating a simple mathematical model to compute a solution and developing computational techniques to solve the model. Each of these issues are studied in turn in this thesis. First, the current optimal face identification models are extended to bounded variable linear programming problems. Distance to the lower and upper bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of projection attempts needed to find an exact solution. When early termination is the goal, projection from a pure composite Newton step is advocated. However, the cost may exceed the benefits due to the average need of more than one projection attempt to find an exact solution. Variants of Mehrotra's predictor-corrector primal-dual interior-point algorithm provide the foundation for most practical interior-point codes. To take advantage of all available algorithmic information, we investigate the behavior of the Tapia predictor-corrector indicator, which incorporates the corrector step, to identify the optimal partition. Globally, the Tapia predictor-corrector indicator behaves poorly as do all indicators, but locally exhibits fast convergence.
Mathematics; Operations research